Soft Active Materials
I. Dielectric Elastomers
Zhigang Suo
Harvard University
1
Hard Machines
2
C-3PO
R2-D2
Soft Machines
Angelina Jolie
Octopus
3
octopus
Mäthger, Hanlon, Kuzirian – Marine Biological Laboratory, Woods Hole MA4
Squid changes color
Expand pigmented sacs by contracting muscles
5
Mathger, Denton, Marshall, Hanlon, J. R. Soc. Interface 6, S149 (2009)
Elastomeric Optics
George Whitesides
•Many stimuli cause deformation.
•Deformation affects optics.
Wilbur, Jackman, Whitesides, Cheung, Lee, Prentiss
Chem. Mater. 8, 1380 (1996)
Aschwanden, Stemmer
Optics letters 31, 2610 (2006)
6
elastomer = network
gel = network + solvent
solvent
reversible
gel
network
7
Super absorbent diaper
Sodium polyacrylate
Masuda, Trends in the development of superabsorbent polymers for diapers, pp. 88-89,
Superabsorbent polymers: science and technology (1994).
8
Oil Spill…Hair
•Hair adsorbs oil (~3g oil/1g hair)
•Polypropylene fibers absorb oil (~10g oil/1g polypropylene)
Phil McCrory, Inventor of the hairmats
Lisa Gautier, Founder of Matter of Trust
Swelling packers in oil wells
Cai, Lou, Ganguly, Robisson, Suo
Forces generated by a swelling elastomer subject to constraint
Journal of Applied Physics, in press
10
Gels regulate flow in plants
Missy Holbrook
Zwieniecki, Melcher, Holbrook,
Hydrogel control of xylem hydraulic resistance in plants
Science 291, 1095 (2001)
11
Self-regulating fluidics
David Beebe
Responsive to
Physiological variables:
•pH
•Salt
•Temperature
•light
•Many stimuli cause deformation.
•Deformation regulates flow.
12
Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000)
Soft Active Materials
Soft: capable of large and reversible deformation
(rubbers, gels,…)
Active: response to diverse stimuli
(electric field, temperature, pH, salt,…)
A stimulus causes deformation.
Stimuli
pH, E, T, C…
Deformation provides a function.
SAM
deforms
Functions
optics, flow…
Better life through deformation
13
Dielectric elastomer
Reference State
A
Current State
Dielectric
Elastomer
L
a
l
Compliant
Electrode
Pelrine, Kornbluh, Pei, Joseph
High-speed electrically actuated elastomers with strain greater than 100%.
Science 287, 836 (2000).
Q

Q
•Large strain
•Noise-less
•Cheap
14
Potential applications of dielectric elastomers
•
Automobil:
In- outside, Temp., Safety, Vandalism, Stupidity, …
•
Aircraft:
In- outside, Temp., Safety, Reliability, Failsave, …
•
Space:
Temp., Vacuum, Radiation, Reliability, Vibration, …
•
Machine Tool:
Vibration, Act-Speed, Positioning, Force, …
•
Robot./Prosthetics:
Safety, Act-Speed, Efficiency, Low Voltage, Force, …
•
Tectile Displays:
Safety, Low Voltage, Multifunction, Resolution, …
•
Micro Actuation:
Low Voltage, Small Size, Fabrication, …
•
FFD:
Multifunction, Safety, Responsetime, …
•
Optics:
Refractivity, Transmittanz, No Imperfections, …
•
Energy Harvesting:
Efficiency, Conductivity, Fatigue, …
•
Sensing:
Low Stiffness, No Viscosity, …
•
Implants:
Safety, Reliability, Very Low Voltage, Radio Control, …
Gabor Kovacs, Winter School , Ascona, Switzerland, 10-16 January 2010
Roll to Roll Manufacturing
Danfoss PolyPower
Roll coating
of film
Metallise
electrodes
Laminate
metallised film
Roll actuators
16
Parallel-plate capacitor
P
battery
a

Q
electrode
l
Q
vacuum
electrode
P
Electric field
Electric displacement field
stress field
force
E

l
Q
D
a
P

a
D  0E
0, permittivity of vacuum
1
2
   0 E 2 Maxwell stress
17
Trouble with Maxwell stress in dielectrics
-----------
+++++++++
Maxwell stress

 33   E 2
2
-----------+
-
+++++++++
Electrostriction
Our complaints:
•In general,  varies with deformation.
•In general, E2 dependence has no special significance.
•Wrong sign?
18
An atom in an electric field
------------
p
e
p
e

battery
Hydrogen atom
+++++++++
External electric field displaces positive and negative charges somewhat.
•Polarization: Induce more charge on the electrodes.
•Deformation: Distort the shape of the electron cloud.
19
A dipole in an electric field
------------

battery
Polar molecules
+++++++++
External electric field reorients dipoles.
•Polarization: Induce more charge on the electrodes.
•Deformation: Distort the shape of the sample.
20
Field equations in vacuum, Maxwell (1873)

Ei  
x i
Electrostatic field
E i
q

x i  0
Fi  qEi
A field of forces maintain equilibrium of a field of charges

Fi 
x j
Q
0



E
E

E
E

 0 j i
k k ij 
2


P

E
Q
0
2
E2
 ij   0 E j E i 
Maxwell stress
P
0
2
E k E k ij
21
Include Maxwell stress in force balance

1
0E2
2
“Free-body” diagram
h
gh
1
gh     0 E 2
2
1 2
E
2
22
James Clerk Maxwell (1831-1879)
“I have not been able to make the next step, namely, to
account by mechanical considerations for these stresses in
the dielectric. I therefore leave the theory at this point…”
A Treatise on Electricity & Magnetism (1873), Article 111
23
Trouble with electric force in dielectrics
In a vacuum,
external force is needed to maintain equilibrium of charges
+Q
+Q
P
Fi  qE i
P
In a solid dielectric,
force between charges is NOT an operational concept
+Q
+Q
Fi  qE i
24
The Feynman Lectures on Physics
Volume II, p.10-8 (1964)
“It is a difficult matter, generally speaking, to make a unique
distinction between the electrical forces and mechanical
forces due to solid material itself. Fortunately, no one ever
really needs to know the answer to the question proposed.
He may sometimes want to know how much strain there is
going to be in a solid, and that can be worked out. But it is
much more complicated than the simple result we got for
liquids.”
25
All troubles are gone if we use measurable quantities
Current State
Reference State
A
a
L
l
equilibrate elastomer and loads
Pl Q


AL AL LA
name quantities
equations of state
~ ~
W  s  ED
 
~
W  , D
s

Suo, Zhao, Greene, J. Mech. Phys. Solids 56, 467 (2008)
 
~
~ W  , D
E
~
D
Q
Q
P
F  Pl  Q
F
divide by volume

Nominal
W  F / AL 
  l/L
True
s  P/A
  P /a
~
E  /L
E   /l
~
D  Q/ A
D  Q/a
26
The nominal vs. the true
Reference State
Current State
A
L
Q
a
l

~
E /L
~
D Q/ A
(E   / l)
( D  Q / a)
Q
P
Nominal electric field and nominal electric displacement are work-conjugate
Battery does work
  
~
~
~ ~
Q  EL  DA   AL ED
True electric field and true electric displacement are NOT work-conjugate
Battery does work
Q  El  Da   la ED  EDl a
27
Dielectric constant is insensitive to stretch
VHB 4910
Kofod, Sommer-Larsen, Kornbluh, Pelrine
Journal of Intelligent Material Systems and Structures 14, 787 (2003).
28
Ideal dielectric elastomer
Dielectric behavior is liquid-like, unaffected by deformation.
 
~
D2
W  , D  Wstretch   
2
Elasticity
Polarization
incompressibility
A
A 1  a
1

a
For an ideal dielectric elastomer, electromechanical coupling is purely a geometric effect:
D
Q
a
~ Q
D
A
~
D  D
~
~
D 2 2
W  , D  Wstretch   
2
 
29
Ideal dielectric elastomer
~
~
D 2 2
W  , D  Wstretch   
2
 
In terms of nominal quantities
 
~
W  , D
s

~2
D 
dWstretch  
s

d

In terms of true quantities
~
D  D
  s
D  E
dWstretch  
 
 E 2
d
30
Maxwell stress represented in three ways
Reference State
A
Current State
Dielectric
Elastomer
L
a
l
Compliant
Electrode
Uniaxial stress
  E 2
biaxial stress
  E 2

Q
Q
triaxial stress
1
2
  E 2
31
For incompressible material, the 3 states of stress give the same state of deformation
Stress-stretch curve
H
1
Elastomer is incompressible
H  h2
1


h



  



32
Deformation of actuation
1
Maxwell stress
H
1
  E 2
voltage   Eh  EH2

Q
h
Q

Equation of state
  H2
  


33
Experimentally observed
deformation of actuation
•
•
•
•
•
•
Ceramics, <1%.
Zhenyi, et al. (1994), polymer ~3%.
Zhang, et al. (1998),polymer ~7%.
Pelrine, et al. (1998), low modulus, high dielectric strength, ~30%.
Pelrine, et al. (2000), pre-stress, ~100%.
Ha, et al. (2006), interpenetrating networks, ~100%.
•What is the theoretical limit?
•How about 1000%?
34
Zhao and Suo, PRL 104, 178302 (2010)
Two modes of failure

Q
l
Q
Pull-in instability
Soft material
Electrical breakdown
Hard material
polarizing
thinning
polarizing


Q
Q
35
Stark & Garton, Nature 176, 1225 (1955)
Electrical breakdown
B
 B  EB h  E B H2

Kofod,
Plante…
To measure dielectric strength,
one must suppress electromechanical instability by fixing stretch.
36
Pull-in instability
limits deformation of actuation
  H2
  

Linear elastic model
  Y   1 
Xuanhe Zhao

c  1.3
c

Experiment: Pelrine, Kornbluh, Joseph,
Sensors & Actuators A 64, 77 (1998)
Calculation: Zhao & Suo
Appl. Phys Lett. 91, 061921 (2007)
37
stiffening
Coexistent states
thinning
polarizing
Q
l


Q
thick
thin
Q
Top view
Cross section
Coexistent states: flat and wrinkled
Observation: Plante, Dubowsky,
Int. J. Solids and Structures 43, 7727 (2006)
Interpretation: Zhao, Hong, Suo
Physical Review B 76, 134113 (2007)
38
Three types of behavior
39
Zhao and Suo, PRL 104, 178302 (2010).
Experimentally observed
deformation of actuation
•
•
•
•
•
•
Ceramics, <1%. Type I
Zhenyi, et al. (1994), polymer ~3%. Type I
Zhang, et al. (1998),polymer ~7%. Type I
Pelrine, et al. (1998), ~30%. Type II
Pelrine, et al. (2000), pre-stress, ~100%. Type III
Ha, et al. (2006), interpenetrating networks, ~100%. Type III
40
Zhao and Suo, PRL 104, 178302 (2010)
Interpenetrating networks
Ha, Yuan, Pei, Pelrine, Adv. Mater. 18, 887 (2006).
Suo, Zhu. APL 95, 232909 (2009).
Zhao and Suo, PRL 104, 178302 (2010).
41
When stretched, a polymer stiffens
n links
released
b
Langevin
b n
P
Gauss
stretched
P

P
n
fully stretched
P
bn
P
42
Arruda-Boyce model
1 23  1



kT
nv
1 2
1  22  23 1 / 2

 1
1
  n 
 
 tanh  

lim ~ n
W
 1  1
W 1 , 2 
 E 2
1
Kuhn, Grun, Kolloid-Z. 101, 248 (1942)
Arruda, Boyce, J. Mech. Phys. Solids 41, 389 (1993)
kT  


 log
v  tanh
sinh 
 2  2



W 1 , 2 
 E 2
2
43
Change contour length of polymer chain
n=5

H
v
kT
B
50
500
44
Effect of Prestress
P
P

1
H
Q
h
1

Q

2
P

     4    
H
H
n = 50
45
Desired stress-stretch curve

hard
Ear lobe
soft

•Soft: enable deformation.
•Hard: avert excessive deformation.
46
Design materials for
desirable stress-stretch curve
47
Dielectric gel

0


2
1
 22  23 
1/2
1 23  1
 1
1
  n 
 
 tanh  
W
Kuhn, Grun, Kolloid-Z. 101, 248 (1942)
Arruda, Boyce, J. Mech. Phys. Solids 41, 389 (1993)
Zhao and Suo, PRL 104, 178302 (2010).
kT  



log
20 v  tanh
sinh 



 1  1
W 1 , 2 
 E 2
1
 2  2
W 1 , 2 
 E 2
2
48
Dielectric gel
49
Zhao and Suo, PRL 104, 178302 (2010).
Energy harvesting
Generate electricity from walking
Generate electricity from waves
50
Pelrine, Kornbluh…
Maximal energy that can be converted
by a dielectric elastomer
1 J/g, Elastomer
1 mJ/g ceramics
E=0
Adrian Koh
EMI
R
•Cheap
•Reliable
EB
LT
51
Koh, Zhao, Suo, Appl. Phys. Lett. 94, 262902 (2009)
VHB and Natural Rubber
(a)
Koh, Keplinger , Li, Bauer, Suo, submitted (2010)
(b)
52
VHB and Natural Rubber
53
Koh, Keplinger , Lii, Bauer, Suo, submitted (2010)
Generator X
Natural Rubber
2
E EB
 0.024MPa
  0.3MPa
54
Koh, Keplinger , Lii, Bauer, Suo, submitted (2010)
Electrostriction
-----------
+++++++++
Maxwell stress

 33   E 2
2
-----------+
-
+++++++++
Electrostriction
55
Non-ideal dielectric elastomer
Dielectric constant
   1  c 1  2  3  3
Area ratio
deformation affects dielectric constant
56
Wissler, Mazza, Sens. Actuators, A 138, 384 (2007).
Quasi-linear dielectric elastomer
    
 
~
D2
W  , D  Wstretch   
2  
 
~
W  , D
s

~2
dWstretch   D d  2 
s



d
2 d    
Zhao, Suo, JAP 104, 123530 (2008)
57
A field of markers: stretch

l
L
Reference state
X
l
L
Current state
x(X, t)
X+dX
x(X+dX, t)
FiK
x i X ,t 

X K
x i X  dX,t   x i X,t 
dX K
58
A field of batteries: electric field
Reference state
L
Current state
l
X
x(X, t)

~ 
E
L
x(X+dX, t)
X+dX
X,t 
X  dX,t 
X  dX ,t   X ,t 
dX K
ground
 X ,t 
~
EK  
X K
59
3D inhomogeneous field
Condition of thermodynamic equilibrium


 2 xi
WdV   Bi   2
t



x i dV  Tix i dA  qdV  dA




Need to specify a material model
 
~
W , D
 ~

~
W F, D
~

~
W F, D  siK FiK  E K DK
Q

siK

~
W F, D

FiK

 
~
W F, D
~
EK 
~
DK
P
l
60
PDEs
FiK
 X ,t 
~
EK  
X K
x X ,t 
 i
X K
siK X, t 
 x i X, t 
 Bi X, t   
X K
t 2
2
siK

~
W F, D

FiK

~
DK X , t 
 qX , t 
X K
 
~
W F, D
~
EK 
~
DK
Toupin (1956), Eringen (1963), Tiersten (1971),
Goulbourne, Mockensturm and Frecker (2005), Dorfmann & Ogden (2005), McMeeking & Landis (2005)…
61
Suo, Zhao, Greene, J. Mech. Phys. Solids 56, 467 (2008)
Finite element method
x i X , t   X , t 
Solve for
FiK 
 X ,t 
~
EK  
X K
x i X ,t 
X K
 
~
~ ~
ˆ
W F, E  W  D K E K
Legendre transformation
Conditions of thermodynamic equilibrium

ˆ 

 2 xi
W
i
dV   Bi  2
FiK X K
t



 i dV  Ti i dA



ˆ 
W
dV  qdV   dA
~
E K X K


62
Zhao, ABAQUS user-supplied subroutine, http://imechanica.org/node/4234
States of equilibrium
P B
 
P
A
L
R

Simple Layer
Zhao and Suo, APL 93, 251902 (2008)
Ring
Sphere
63
Oscillating Balloon
Excitation by sinusoidal voltage
Oscillation of the balloon
Zhao and Suo, APL 93, 251902 (2008)
Zhu, Cai, Suo, Polymer International 59, 378-383 (2010)
64
Programmable deformation
Design:
Kofod, Wirges, Paajanen, Bauer
APL 90, 081916 (2007)
Simulation:
Zhao, Suo
APL 93, 251902 (2008)
65
Summary
• Soft active materials (SAMs) have many functions (soft robots, adaptive
optics, self-regulating fluidics, programmable haptics, oil recovery, energy
harvesting, drug delivery, tissue regeneration, low-cost diagnosis, oil spill
cleanup…)
• SAMs is interesting and challenging to study (mechanics, electrostatics,
chemistry; large deformation, mass transport, instability).
• The field is wide open (stimuli, SAMs, functions).
Stimuli
pH, E, T, C…
SAM
deforms
Functions
optics, flow…
66